1 /* 2 * Copyright 2006 The Android Open Source Project 3 * 4 * Use of this source code is governed by a BSD-style license that can be 5 * found in the LICENSE file. 6 */ 7 8 #include "SkGeometry.h" 9 #include "SkMatrix.h" 10 11 bool SkXRayCrossesLine(const SkXRay& pt, 12 const SkPoint pts[2], 13 bool* ambiguous) { 14 if (ambiguous) { 15 *ambiguous = false; 16 } 17 // Determine quick discards. 18 // Consider query line going exactly through point 0 to not 19 // intersect, for symmetry with SkXRayCrossesMonotonicCubic. 20 if (pt.fY == pts[0].fY) { 21 if (ambiguous) { 22 *ambiguous = true; 23 } 24 return false; 25 } 26 if (pt.fY < pts[0].fY && pt.fY < pts[1].fY) 27 return false; 28 if (pt.fY > pts[0].fY && pt.fY > pts[1].fY) 29 return false; 30 if (pt.fX > pts[0].fX && pt.fX > pts[1].fX) 31 return false; 32 // Determine degenerate cases 33 if (SkScalarNearlyZero(pts[0].fY - pts[1].fY)) 34 return false; 35 if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) { 36 // We've already determined the query point lies within the 37 // vertical range of the line segment. 38 if (pt.fX <= pts[0].fX) { 39 if (ambiguous) { 40 *ambiguous = (pt.fY == pts[1].fY); 41 } 42 return true; 43 } 44 return false; 45 } 46 // Ambiguity check 47 if (pt.fY == pts[1].fY) { 48 if (pt.fX <= pts[1].fX) { 49 if (ambiguous) { 50 *ambiguous = true; 51 } 52 return true; 53 } 54 return false; 55 } 56 // Full line segment evaluation 57 SkScalar delta_y = pts[1].fY - pts[0].fY; 58 SkScalar delta_x = pts[1].fX - pts[0].fX; 59 SkScalar slope = SkScalarDiv(delta_y, delta_x); 60 SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX); 61 // Solve for x coordinate at y = pt.fY 62 SkScalar x = SkScalarDiv(pt.fY - b, slope); 63 return pt.fX <= x; 64 } 65 66 /** If defined, this makes eval_quad and eval_cubic do more setup (sometimes 67 involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. 68 May also introduce overflow of fixed when we compute our setup. 69 */ 70 // #define DIRECT_EVAL_OF_POLYNOMIALS 71 72 //////////////////////////////////////////////////////////////////////// 73 74 static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { 75 SkScalar ab = a - b; 76 SkScalar bc = b - c; 77 if (ab < 0) { 78 bc = -bc; 79 } 80 return ab == 0 || bc < 0; 81 } 82 83 //////////////////////////////////////////////////////////////////////// 84 85 static bool is_unit_interval(SkScalar x) { 86 return x > 0 && x < SK_Scalar1; 87 } 88 89 static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { 90 SkASSERT(ratio); 91 92 if (numer < 0) { 93 numer = -numer; 94 denom = -denom; 95 } 96 97 if (denom == 0 || numer == 0 || numer >= denom) { 98 return 0; 99 } 100 101 SkScalar r = SkScalarDiv(numer, denom); 102 if (SkScalarIsNaN(r)) { 103 return 0; 104 } 105 SkASSERT(r >= 0 && r < SK_Scalar1); 106 if (r == 0) { // catch underflow if numer <<<< denom 107 return 0; 108 } 109 *ratio = r; 110 return 1; 111 } 112 113 /** From Numerical Recipes in C. 114 115 Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) 116 x1 = Q / A 117 x2 = C / Q 118 */ 119 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { 120 SkASSERT(roots); 121 122 if (A == 0) { 123 return valid_unit_divide(-C, B, roots); 124 } 125 126 SkScalar* r = roots; 127 128 SkScalar R = B*B - 4*A*C; 129 if (R < 0 || SkScalarIsNaN(R)) { // complex roots 130 return 0; 131 } 132 R = SkScalarSqrt(R); 133 134 SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; 135 r += valid_unit_divide(Q, A, r); 136 r += valid_unit_divide(C, Q, r); 137 if (r - roots == 2) { 138 if (roots[0] > roots[1]) 139 SkTSwap<SkScalar>(roots[0], roots[1]); 140 else if (roots[0] == roots[1]) // nearly-equal? 141 r -= 1; // skip the double root 142 } 143 return (int)(r - roots); 144 } 145 146 /////////////////////////////////////////////////////////////////////////////// 147 /////////////////////////////////////////////////////////////////////////////// 148 149 static SkScalar eval_quad(const SkScalar src[], SkScalar t) { 150 SkASSERT(src); 151 SkASSERT(t >= 0 && t <= SK_Scalar1); 152 153 #ifdef DIRECT_EVAL_OF_POLYNOMIALS 154 SkScalar C = src[0]; 155 SkScalar A = src[4] - 2 * src[2] + C; 156 SkScalar B = 2 * (src[2] - C); 157 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 158 #else 159 SkScalar ab = SkScalarInterp(src[0], src[2], t); 160 SkScalar bc = SkScalarInterp(src[2], src[4], t); 161 return SkScalarInterp(ab, bc, t); 162 #endif 163 } 164 165 static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) { 166 SkScalar A = src[4] - 2 * src[2] + src[0]; 167 SkScalar B = src[2] - src[0]; 168 169 return 2 * SkScalarMulAdd(A, t, B); 170 } 171 172 static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) { 173 SkScalar A = src[4] - 2 * src[2] + src[0]; 174 SkScalar B = src[2] - src[0]; 175 return A + 2 * B; 176 } 177 178 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, 179 SkVector* tangent) { 180 SkASSERT(src); 181 SkASSERT(t >= 0 && t <= SK_Scalar1); 182 183 if (pt) { 184 pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); 185 } 186 if (tangent) { 187 tangent->set(eval_quad_derivative(&src[0].fX, t), 188 eval_quad_derivative(&src[0].fY, t)); 189 } 190 } 191 192 void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) { 193 SkASSERT(src); 194 195 if (pt) { 196 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 197 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 198 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 199 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 200 pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 201 } 202 if (tangent) { 203 tangent->set(eval_quad_derivative_at_half(&src[0].fX), 204 eval_quad_derivative_at_half(&src[0].fY)); 205 } 206 } 207 208 static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) { 209 SkScalar ab = SkScalarInterp(src[0], src[2], t); 210 SkScalar bc = SkScalarInterp(src[2], src[4], t); 211 212 dst[0] = src[0]; 213 dst[2] = ab; 214 dst[4] = SkScalarInterp(ab, bc, t); 215 dst[6] = bc; 216 dst[8] = src[4]; 217 } 218 219 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { 220 SkASSERT(t > 0 && t < SK_Scalar1); 221 222 interp_quad_coords(&src[0].fX, &dst[0].fX, t); 223 interp_quad_coords(&src[0].fY, &dst[0].fY, t); 224 } 225 226 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { 227 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 228 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 229 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 230 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 231 232 dst[0] = src[0]; 233 dst[1].set(x01, y01); 234 dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); 235 dst[3].set(x12, y12); 236 dst[4] = src[2]; 237 } 238 239 /** Quad'(t) = At + B, where 240 A = 2(a - 2b + c) 241 B = 2(b - a) 242 Solve for t, only if it fits between 0 < t < 1 243 */ 244 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { 245 /* At + B == 0 246 t = -B / A 247 */ 248 return valid_unit_divide(a - b, a - b - b + c, tValue); 249 } 250 251 static inline void flatten_double_quad_extrema(SkScalar coords[14]) { 252 coords[2] = coords[6] = coords[4]; 253 } 254 255 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 256 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 257 */ 258 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { 259 SkASSERT(src); 260 SkASSERT(dst); 261 262 SkScalar a = src[0].fY; 263 SkScalar b = src[1].fY; 264 SkScalar c = src[2].fY; 265 266 if (is_not_monotonic(a, b, c)) { 267 SkScalar tValue; 268 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 269 SkChopQuadAt(src, dst, tValue); 270 flatten_double_quad_extrema(&dst[0].fY); 271 return 1; 272 } 273 // if we get here, we need to force dst to be monotonic, even though 274 // we couldn't compute a unit_divide value (probably underflow). 275 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 276 } 277 dst[0].set(src[0].fX, a); 278 dst[1].set(src[1].fX, b); 279 dst[2].set(src[2].fX, c); 280 return 0; 281 } 282 283 /* Returns 0 for 1 quad, and 1 for two quads, either way the answer is 284 stored in dst[]. Guarantees that the 1/2 quads will be monotonic. 285 */ 286 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { 287 SkASSERT(src); 288 SkASSERT(dst); 289 290 SkScalar a = src[0].fX; 291 SkScalar b = src[1].fX; 292 SkScalar c = src[2].fX; 293 294 if (is_not_monotonic(a, b, c)) { 295 SkScalar tValue; 296 if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { 297 SkChopQuadAt(src, dst, tValue); 298 flatten_double_quad_extrema(&dst[0].fX); 299 return 1; 300 } 301 // if we get here, we need to force dst to be monotonic, even though 302 // we couldn't compute a unit_divide value (probably underflow). 303 b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; 304 } 305 dst[0].set(a, src[0].fY); 306 dst[1].set(b, src[1].fY); 307 dst[2].set(c, src[2].fY); 308 return 0; 309 } 310 311 // F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 312 // F'(t) = 2 (b - a) + 2 (a - 2b + c) t 313 // F''(t) = 2 (a - 2b + c) 314 // 315 // A = 2 (b - a) 316 // B = 2 (a - 2b + c) 317 // 318 // Maximum curvature for a quadratic means solving 319 // Fx' Fx'' + Fy' Fy'' = 0 320 // 321 // t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) 322 // 323 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { 324 SkScalar Ax = src[1].fX - src[0].fX; 325 SkScalar Ay = src[1].fY - src[0].fY; 326 SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; 327 SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; 328 SkScalar t = 0; // 0 means don't chop 329 330 (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); 331 return t; 332 } 333 334 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { 335 SkScalar t = SkFindQuadMaxCurvature(src); 336 if (t == 0) { 337 memcpy(dst, src, 3 * sizeof(SkPoint)); 338 return 1; 339 } else { 340 SkChopQuadAt(src, dst, t); 341 return 2; 342 } 343 } 344 345 #define SK_ScalarTwoThirds (0.666666666f) 346 347 void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { 348 const SkScalar scale = SK_ScalarTwoThirds; 349 dst[0] = src[0]; 350 dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), 351 src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); 352 dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), 353 src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); 354 dst[3] = src[2]; 355 } 356 357 ////////////////////////////////////////////////////////////////////////////// 358 ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// 359 ////////////////////////////////////////////////////////////////////////////// 360 361 static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) { 362 coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; 363 coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); 364 coeff[2] = 3*(pt[2] - pt[0]); 365 coeff[3] = pt[0]; 366 } 367 368 void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) { 369 SkASSERT(pts); 370 371 if (cx) { 372 get_cubic_coeff(&pts[0].fX, cx); 373 } 374 if (cy) { 375 get_cubic_coeff(&pts[0].fY, cy); 376 } 377 } 378 379 static SkScalar eval_cubic(const SkScalar src[], SkScalar t) { 380 SkASSERT(src); 381 SkASSERT(t >= 0 && t <= SK_Scalar1); 382 383 if (t == 0) { 384 return src[0]; 385 } 386 387 #ifdef DIRECT_EVAL_OF_POLYNOMIALS 388 SkScalar D = src[0]; 389 SkScalar A = src[6] + 3*(src[2] - src[4]) - D; 390 SkScalar B = 3*(src[4] - src[2] - src[2] + D); 391 SkScalar C = 3*(src[2] - D); 392 393 return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); 394 #else 395 SkScalar ab = SkScalarInterp(src[0], src[2], t); 396 SkScalar bc = SkScalarInterp(src[2], src[4], t); 397 SkScalar cd = SkScalarInterp(src[4], src[6], t); 398 SkScalar abc = SkScalarInterp(ab, bc, t); 399 SkScalar bcd = SkScalarInterp(bc, cd, t); 400 return SkScalarInterp(abc, bcd, t); 401 #endif 402 } 403 404 /** return At^2 + Bt + C 405 */ 406 static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) { 407 SkASSERT(t >= 0 && t <= SK_Scalar1); 408 409 return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 410 } 411 412 static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) { 413 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 414 SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); 415 SkScalar C = src[2] - src[0]; 416 417 return eval_quadratic(A, B, C, t); 418 } 419 420 static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) { 421 SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; 422 SkScalar B = src[4] - 2 * src[2] + src[0]; 423 424 return SkScalarMulAdd(A, t, B); 425 } 426 427 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, 428 SkVector* tangent, SkVector* curvature) { 429 SkASSERT(src); 430 SkASSERT(t >= 0 && t <= SK_Scalar1); 431 432 if (loc) { 433 loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); 434 } 435 if (tangent) { 436 tangent->set(eval_cubic_derivative(&src[0].fX, t), 437 eval_cubic_derivative(&src[0].fY, t)); 438 } 439 if (curvature) { 440 curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), 441 eval_cubic_2ndDerivative(&src[0].fY, t)); 442 } 443 } 444 445 /** Cubic'(t) = At^2 + Bt + C, where 446 A = 3(-a + 3(b - c) + d) 447 B = 6(a - 2b + c) 448 C = 3(b - a) 449 Solve for t, keeping only those that fit betwee 0 < t < 1 450 */ 451 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, 452 SkScalar tValues[2]) { 453 // we divide A,B,C by 3 to simplify 454 SkScalar A = d - a + 3*(b - c); 455 SkScalar B = 2*(a - b - b + c); 456 SkScalar C = b - a; 457 458 return SkFindUnitQuadRoots(A, B, C, tValues); 459 } 460 461 static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, 462 SkScalar t) { 463 SkScalar ab = SkScalarInterp(src[0], src[2], t); 464 SkScalar bc = SkScalarInterp(src[2], src[4], t); 465 SkScalar cd = SkScalarInterp(src[4], src[6], t); 466 SkScalar abc = SkScalarInterp(ab, bc, t); 467 SkScalar bcd = SkScalarInterp(bc, cd, t); 468 SkScalar abcd = SkScalarInterp(abc, bcd, t); 469 470 dst[0] = src[0]; 471 dst[2] = ab; 472 dst[4] = abc; 473 dst[6] = abcd; 474 dst[8] = bcd; 475 dst[10] = cd; 476 dst[12] = src[6]; 477 } 478 479 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { 480 SkASSERT(t > 0 && t < SK_Scalar1); 481 482 interp_cubic_coords(&src[0].fX, &dst[0].fX, t); 483 interp_cubic_coords(&src[0].fY, &dst[0].fY, t); 484 } 485 486 /* http://code.google.com/p/skia/issues/detail?id=32 487 488 This test code would fail when we didn't check the return result of 489 valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is 490 that after the first chop, the parameters to valid_unit_divide are equal 491 (thanks to finite float precision and rounding in the subtracts). Thus 492 even though the 2nd tValue looks < 1.0, after we renormalize it, we end 493 up with 1.0, hence the need to check and just return the last cubic as 494 a degenerate clump of 4 points in the sampe place. 495 496 static void test_cubic() { 497 SkPoint src[4] = { 498 { 556.25000, 523.03003 }, 499 { 556.23999, 522.96002 }, 500 { 556.21997, 522.89001 }, 501 { 556.21997, 522.82001 } 502 }; 503 SkPoint dst[10]; 504 SkScalar tval[] = { 0.33333334f, 0.99999994f }; 505 SkChopCubicAt(src, dst, tval, 2); 506 } 507 */ 508 509 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], 510 const SkScalar tValues[], int roots) { 511 #ifdef SK_DEBUG 512 { 513 for (int i = 0; i < roots - 1; i++) 514 { 515 SkASSERT(is_unit_interval(tValues[i])); 516 SkASSERT(is_unit_interval(tValues[i+1])); 517 SkASSERT(tValues[i] < tValues[i+1]); 518 } 519 } 520 #endif 521 522 if (dst) { 523 if (roots == 0) { // nothing to chop 524 memcpy(dst, src, 4*sizeof(SkPoint)); 525 } else { 526 SkScalar t = tValues[0]; 527 SkPoint tmp[4]; 528 529 for (int i = 0; i < roots; i++) { 530 SkChopCubicAt(src, dst, t); 531 if (i == roots - 1) { 532 break; 533 } 534 535 dst += 3; 536 // have src point to the remaining cubic (after the chop) 537 memcpy(tmp, dst, 4 * sizeof(SkPoint)); 538 src = tmp; 539 540 // watch out in case the renormalized t isn't in range 541 if (!valid_unit_divide(tValues[i+1] - tValues[i], 542 SK_Scalar1 - tValues[i], &t)) { 543 // if we can't, just create a degenerate cubic 544 dst[4] = dst[5] = dst[6] = src[3]; 545 break; 546 } 547 } 548 } 549 } 550 } 551 552 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { 553 SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); 554 SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); 555 SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); 556 SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); 557 SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); 558 SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); 559 560 SkScalar x012 = SkScalarAve(x01, x12); 561 SkScalar y012 = SkScalarAve(y01, y12); 562 SkScalar x123 = SkScalarAve(x12, x23); 563 SkScalar y123 = SkScalarAve(y12, y23); 564 565 dst[0] = src[0]; 566 dst[1].set(x01, y01); 567 dst[2].set(x012, y012); 568 dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); 569 dst[4].set(x123, y123); 570 dst[5].set(x23, y23); 571 dst[6] = src[3]; 572 } 573 574 static void flatten_double_cubic_extrema(SkScalar coords[14]) { 575 coords[4] = coords[8] = coords[6]; 576 } 577 578 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that 579 the resulting beziers are monotonic in Y. This is called by the scan 580 converter. Depending on what is returned, dst[] is treated as follows: 581 0 dst[0..3] is the original cubic 582 1 dst[0..3] and dst[3..6] are the two new cubics 583 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics 584 If dst == null, it is ignored and only the count is returned. 585 */ 586 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { 587 SkScalar tValues[2]; 588 int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, 589 src[3].fY, tValues); 590 591 SkChopCubicAt(src, dst, tValues, roots); 592 if (dst && roots > 0) { 593 // we do some cleanup to ensure our Y extrema are flat 594 flatten_double_cubic_extrema(&dst[0].fY); 595 if (roots == 2) { 596 flatten_double_cubic_extrema(&dst[3].fY); 597 } 598 } 599 return roots; 600 } 601 602 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { 603 SkScalar tValues[2]; 604 int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, 605 src[3].fX, tValues); 606 607 SkChopCubicAt(src, dst, tValues, roots); 608 if (dst && roots > 0) { 609 // we do some cleanup to ensure our Y extrema are flat 610 flatten_double_cubic_extrema(&dst[0].fX); 611 if (roots == 2) { 612 flatten_double_cubic_extrema(&dst[3].fX); 613 } 614 } 615 return roots; 616 } 617 618 /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html 619 620 Inflection means that curvature is zero. 621 Curvature is [F' x F''] / [F'^3] 622 So we solve F'x X F''y - F'y X F''y == 0 623 After some canceling of the cubic term, we get 624 A = b - a 625 B = c - 2b + a 626 C = d - 3c + 3b - a 627 (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 628 */ 629 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { 630 SkScalar Ax = src[1].fX - src[0].fX; 631 SkScalar Ay = src[1].fY - src[0].fY; 632 SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; 633 SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; 634 SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; 635 SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; 636 637 return SkFindUnitQuadRoots(Bx*Cy - By*Cx, 638 Ax*Cy - Ay*Cx, 639 Ax*By - Ay*Bx, 640 tValues); 641 } 642 643 int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { 644 SkScalar tValues[2]; 645 int count = SkFindCubicInflections(src, tValues); 646 647 if (dst) { 648 if (count == 0) { 649 memcpy(dst, src, 4 * sizeof(SkPoint)); 650 } else { 651 SkChopCubicAt(src, dst, tValues, count); 652 } 653 } 654 return count + 1; 655 } 656 657 template <typename T> void bubble_sort(T array[], int count) { 658 for (int i = count - 1; i > 0; --i) 659 for (int j = i; j > 0; --j) 660 if (array[j] < array[j-1]) 661 { 662 T tmp(array[j]); 663 array[j] = array[j-1]; 664 array[j-1] = tmp; 665 } 666 } 667 668 /** 669 * Given an array and count, remove all pair-wise duplicates from the array, 670 * keeping the existing sorting, and return the new count 671 */ 672 static int collaps_duplicates(SkScalar array[], int count) { 673 for (int n = count; n > 1; --n) { 674 if (array[0] == array[1]) { 675 for (int i = 1; i < n; ++i) { 676 array[i - 1] = array[i]; 677 } 678 count -= 1; 679 } else { 680 array += 1; 681 } 682 } 683 return count; 684 } 685 686 #ifdef SK_DEBUG 687 688 #define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) 689 690 static void test_collaps_duplicates() { 691 static bool gOnce; 692 if (gOnce) { return; } 693 gOnce = true; 694 const SkScalar src0[] = { 0 }; 695 const SkScalar src1[] = { 0, 0 }; 696 const SkScalar src2[] = { 0, 1 }; 697 const SkScalar src3[] = { 0, 0, 0 }; 698 const SkScalar src4[] = { 0, 0, 1 }; 699 const SkScalar src5[] = { 0, 1, 1 }; 700 const SkScalar src6[] = { 0, 1, 2 }; 701 const struct { 702 const SkScalar* fData; 703 int fCount; 704 int fCollapsedCount; 705 } data[] = { 706 { TEST_COLLAPS_ENTRY(src0), 1 }, 707 { TEST_COLLAPS_ENTRY(src1), 1 }, 708 { TEST_COLLAPS_ENTRY(src2), 2 }, 709 { TEST_COLLAPS_ENTRY(src3), 1 }, 710 { TEST_COLLAPS_ENTRY(src4), 2 }, 711 { TEST_COLLAPS_ENTRY(src5), 2 }, 712 { TEST_COLLAPS_ENTRY(src6), 3 }, 713 }; 714 for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { 715 SkScalar dst[3]; 716 memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); 717 int count = collaps_duplicates(dst, data[i].fCount); 718 SkASSERT(data[i].fCollapsedCount == count); 719 for (int j = 1; j < count; ++j) { 720 SkASSERT(dst[j-1] < dst[j]); 721 } 722 } 723 } 724 #endif 725 726 static SkScalar SkScalarCubeRoot(SkScalar x) { 727 return SkScalarPow(x, 0.3333333f); 728 } 729 730 /* Solve coeff(t) == 0, returning the number of roots that 731 lie withing 0 < t < 1. 732 coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] 733 734 Eliminates repeated roots (so that all tValues are distinct, and are always 735 in increasing order. 736 */ 737 static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { 738 if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic 739 return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); 740 } 741 742 SkScalar a, b, c, Q, R; 743 744 { 745 SkASSERT(coeff[0] != 0); 746 747 SkScalar inva = SkScalarInvert(coeff[0]); 748 a = coeff[1] * inva; 749 b = coeff[2] * inva; 750 c = coeff[3] * inva; 751 } 752 Q = (a*a - b*3) / 9; 753 R = (2*a*a*a - 9*a*b + 27*c) / 54; 754 755 SkScalar Q3 = Q * Q * Q; 756 SkScalar R2MinusQ3 = R * R - Q3; 757 SkScalar adiv3 = a / 3; 758 759 SkScalar* roots = tValues; 760 SkScalar r; 761 762 if (R2MinusQ3 < 0) { // we have 3 real roots 763 SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3)); 764 SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); 765 766 r = neg2RootQ * SkScalarCos(theta/3) - adiv3; 767 if (is_unit_interval(r)) { 768 *roots++ = r; 769 } 770 r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; 771 if (is_unit_interval(r)) { 772 *roots++ = r; 773 } 774 r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; 775 if (is_unit_interval(r)) { 776 *roots++ = r; 777 } 778 SkDEBUGCODE(test_collaps_duplicates();) 779 780 // now sort the roots 781 int count = (int)(roots - tValues); 782 SkASSERT((unsigned)count <= 3); 783 bubble_sort(tValues, count); 784 count = collaps_duplicates(tValues, count); 785 roots = tValues + count; // so we compute the proper count below 786 } else { // we have 1 real root 787 SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); 788 A = SkScalarCubeRoot(A); 789 if (R > 0) { 790 A = -A; 791 } 792 if (A != 0) { 793 A += Q / A; 794 } 795 r = A - adiv3; 796 if (is_unit_interval(r)) { 797 *roots++ = r; 798 } 799 } 800 801 return (int)(roots - tValues); 802 } 803 804 /* Looking for F' dot F'' == 0 805 806 A = b - a 807 B = c - 2b + a 808 C = d - 3c + 3b - a 809 810 F' = 3Ct^2 + 6Bt + 3A 811 F'' = 6Ct + 6B 812 813 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 814 */ 815 static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { 816 SkScalar a = src[2] - src[0]; 817 SkScalar b = src[4] - 2 * src[2] + src[0]; 818 SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; 819 820 coeff[0] = c * c; 821 coeff[1] = 3 * b * c; 822 coeff[2] = 2 * b * b + c * a; 823 coeff[3] = a * b; 824 } 825 826 /* Looking for F' dot F'' == 0 827 828 A = b - a 829 B = c - 2b + a 830 C = d - 3c + 3b - a 831 832 F' = 3Ct^2 + 6Bt + 3A 833 F'' = 6Ct + 6B 834 835 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 836 */ 837 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { 838 SkScalar coeffX[4], coeffY[4]; 839 int i; 840 841 formulate_F1DotF2(&src[0].fX, coeffX); 842 formulate_F1DotF2(&src[0].fY, coeffY); 843 844 for (i = 0; i < 4; i++) { 845 coeffX[i] += coeffY[i]; 846 } 847 848 SkScalar t[3]; 849 int count = solve_cubic_poly(coeffX, t); 850 int maxCount = 0; 851 852 // now remove extrema where the curvature is zero (mins) 853 // !!!! need a test for this !!!! 854 for (i = 0; i < count; i++) { 855 // if (not_min_curvature()) 856 if (t[i] > 0 && t[i] < SK_Scalar1) { 857 tValues[maxCount++] = t[i]; 858 } 859 } 860 return maxCount; 861 } 862 863 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], 864 SkScalar tValues[3]) { 865 SkScalar t_storage[3]; 866 867 if (tValues == NULL) { 868 tValues = t_storage; 869 } 870 871 int count = SkFindCubicMaxCurvature(src, tValues); 872 873 if (dst) { 874 if (count == 0) { 875 memcpy(dst, src, 4 * sizeof(SkPoint)); 876 } else { 877 SkChopCubicAt(src, dst, tValues, count); 878 } 879 } 880 return count + 1; 881 } 882 883 bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], 884 bool* ambiguous) { 885 if (ambiguous) { 886 *ambiguous = false; 887 } 888 889 // Find the minimum and maximum y of the extrema, which are the 890 // first and last points since this cubic is monotonic 891 SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY); 892 SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY); 893 894 if (pt.fY == cubic[0].fY 895 || pt.fY < min_y 896 || pt.fY > max_y) { 897 // The query line definitely does not cross the curve 898 if (ambiguous) { 899 *ambiguous = (pt.fY == cubic[0].fY); 900 } 901 return false; 902 } 903 904 bool pt_at_extremum = (pt.fY == cubic[3].fY); 905 906 SkScalar min_x = 907 SkMinScalar( 908 SkMinScalar( 909 SkMinScalar(cubic[0].fX, cubic[1].fX), 910 cubic[2].fX), 911 cubic[3].fX); 912 if (pt.fX < min_x) { 913 // The query line definitely crosses the curve 914 if (ambiguous) { 915 *ambiguous = pt_at_extremum; 916 } 917 return true; 918 } 919 920 SkScalar max_x = 921 SkMaxScalar( 922 SkMaxScalar( 923 SkMaxScalar(cubic[0].fX, cubic[1].fX), 924 cubic[2].fX), 925 cubic[3].fX); 926 if (pt.fX > max_x) { 927 // The query line definitely does not cross the curve 928 return false; 929 } 930 931 // Do a binary search to find the parameter value which makes y as 932 // close as possible to the query point. See whether the query 933 // line's origin is to the left of the associated x coordinate. 934 935 // kMaxIter is chosen as the number of mantissa bits for a float, 936 // since there's no way we are going to get more precision by 937 // iterating more times than that. 938 const int kMaxIter = 23; 939 SkPoint eval; 940 int iter = 0; 941 SkScalar upper_t; 942 SkScalar lower_t; 943 // Need to invert direction of t parameter if cubic goes up 944 // instead of down 945 if (cubic[3].fY > cubic[0].fY) { 946 upper_t = SK_Scalar1; 947 lower_t = 0; 948 } else { 949 upper_t = 0; 950 lower_t = SK_Scalar1; 951 } 952 do { 953 SkScalar t = SkScalarAve(upper_t, lower_t); 954 SkEvalCubicAt(cubic, t, &eval, NULL, NULL); 955 if (pt.fY > eval.fY) { 956 lower_t = t; 957 } else { 958 upper_t = t; 959 } 960 } while (++iter < kMaxIter 961 && !SkScalarNearlyZero(eval.fY - pt.fY)); 962 if (pt.fX <= eval.fX) { 963 if (ambiguous) { 964 *ambiguous = pt_at_extremum; 965 } 966 return true; 967 } 968 return false; 969 } 970 971 int SkNumXRayCrossingsForCubic(const SkXRay& pt, 972 const SkPoint cubic[4], 973 bool* ambiguous) { 974 int num_crossings = 0; 975 SkPoint monotonic_cubics[10]; 976 int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics); 977 if (ambiguous) { 978 *ambiguous = false; 979 } 980 bool locally_ambiguous; 981 if (SkXRayCrossesMonotonicCubic(pt, 982 &monotonic_cubics[0], 983 &locally_ambiguous)) 984 ++num_crossings; 985 if (ambiguous) { 986 *ambiguous |= locally_ambiguous; 987 } 988 if (num_monotonic_cubics > 0) 989 if (SkXRayCrossesMonotonicCubic(pt, 990 &monotonic_cubics[3], 991 &locally_ambiguous)) 992 ++num_crossings; 993 if (ambiguous) { 994 *ambiguous |= locally_ambiguous; 995 } 996 if (num_monotonic_cubics > 1) 997 if (SkXRayCrossesMonotonicCubic(pt, 998 &monotonic_cubics[6], 999 &locally_ambiguous)) 1000 ++num_crossings; 1001 if (ambiguous) { 1002 *ambiguous |= locally_ambiguous; 1003 } 1004 return num_crossings; 1005 } 1006 1007 /////////////////////////////////////////////////////////////////////////////// 1008 1009 /* Find t value for quadratic [a, b, c] = d. 1010 Return 0 if there is no solution within [0, 1) 1011 */ 1012 static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) { 1013 // At^2 + Bt + C = d 1014 SkScalar A = a - 2 * b + c; 1015 SkScalar B = 2 * (b - a); 1016 SkScalar C = a - d; 1017 1018 SkScalar roots[2]; 1019 int count = SkFindUnitQuadRoots(A, B, C, roots); 1020 1021 SkASSERT(count <= 1); 1022 return count == 1 ? roots[0] : 0; 1023 } 1024 1025 /* given a quad-curve and a point (x,y), chop the quad at that point and place 1026 the new off-curve point and endpoint into 'dest'. 1027 Should only return false if the computed pos is the start of the curve 1028 (i.e. root == 0) 1029 */ 1030 static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, 1031 SkPoint* dest) { 1032 const SkScalar* base; 1033 SkScalar value; 1034 1035 if (SkScalarAbs(x) < SkScalarAbs(y)) { 1036 base = &quad[0].fX; 1037 value = x; 1038 } else { 1039 base = &quad[0].fY; 1040 value = y; 1041 } 1042 1043 // note: this returns 0 if it thinks value is out of range, meaning the 1044 // root might return something outside of [0, 1) 1045 SkScalar t = quad_solve(base[0], base[2], base[4], value); 1046 1047 if (t > 0) { 1048 SkPoint tmp[5]; 1049 SkChopQuadAt(quad, tmp, t); 1050 dest[0] = tmp[1]; 1051 dest[1].set(x, y); 1052 return true; 1053 } else { 1054 /* t == 0 means either the value triggered a root outside of [0, 1) 1055 For our purposes, we can ignore the <= 0 roots, but we want to 1056 catch the >= 1 roots (which given our caller, will basically mean 1057 a root of 1, give-or-take numerical instability). If we are in the 1058 >= 1 case, return the existing offCurve point. 1059 1060 The test below checks to see if we are close to the "end" of the 1061 curve (near base[4]). Rather than specifying a tolerance, I just 1062 check to see if value is on to the right/left of the middle point 1063 (depending on the direction/sign of the end points). 1064 */ 1065 if ((base[0] < base[4] && value > base[2]) || 1066 (base[0] > base[4] && value < base[2])) // should root have been 1 1067 { 1068 dest[0] = quad[1]; 1069 dest[1].set(x, y); 1070 return true; 1071 } 1072 } 1073 return false; 1074 } 1075 1076 static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { 1077 // The mid point of the quadratic arc approximation is half way between the two 1078 // control points. The float epsilon adjustment moves the on curve point out by 1079 // two bits, distributing the convex test error between the round rect 1080 // approximation and the convex cross product sign equality test. 1081 #define SK_MID_RRECT_OFFSET \ 1082 (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 1083 { SK_Scalar1, 0 }, 1084 { SK_Scalar1, SK_ScalarTanPIOver8 }, 1085 { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1086 { SK_ScalarTanPIOver8, SK_Scalar1 }, 1087 1088 { 0, SK_Scalar1 }, 1089 { -SK_ScalarTanPIOver8, SK_Scalar1 }, 1090 { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, 1091 { -SK_Scalar1, SK_ScalarTanPIOver8 }, 1092 1093 { -SK_Scalar1, 0 }, 1094 { -SK_Scalar1, -SK_ScalarTanPIOver8 }, 1095 { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1096 { -SK_ScalarTanPIOver8, -SK_Scalar1 }, 1097 1098 { 0, -SK_Scalar1 }, 1099 { SK_ScalarTanPIOver8, -SK_Scalar1 }, 1100 { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, 1101 { SK_Scalar1, -SK_ScalarTanPIOver8 }, 1102 1103 { SK_Scalar1, 0 } 1104 #undef SK_MID_RRECT_OFFSET 1105 }; 1106 1107 int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, 1108 SkRotationDirection dir, const SkMatrix* userMatrix, 1109 SkPoint quadPoints[]) { 1110 // rotate by x,y so that uStart is (1.0) 1111 SkScalar x = SkPoint::DotProduct(uStart, uStop); 1112 SkScalar y = SkPoint::CrossProduct(uStart, uStop); 1113 1114 SkScalar absX = SkScalarAbs(x); 1115 SkScalar absY = SkScalarAbs(y); 1116 1117 int pointCount; 1118 1119 // check for (effectively) coincident vectors 1120 // this can happen if our angle is nearly 0 or nearly 180 (y == 0) 1121 // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) 1122 if (absY <= SK_ScalarNearlyZero && x > 0 && 1123 ((y >= 0 && kCW_SkRotationDirection == dir) || 1124 (y <= 0 && kCCW_SkRotationDirection == dir))) { 1125 1126 // just return the start-point 1127 quadPoints[0].set(SK_Scalar1, 0); 1128 pointCount = 1; 1129 } else { 1130 if (dir == kCCW_SkRotationDirection) { 1131 y = -y; 1132 } 1133 // what octant (quadratic curve) is [xy] in? 1134 int oct = 0; 1135 bool sameSign = true; 1136 1137 if (0 == y) { 1138 oct = 4; // 180 1139 SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); 1140 } else if (0 == x) { 1141 SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); 1142 oct = y > 0 ? 2 : 6; // 90 : 270 1143 } else { 1144 if (y < 0) { 1145 oct += 4; 1146 } 1147 if ((x < 0) != (y < 0)) { 1148 oct += 2; 1149 sameSign = false; 1150 } 1151 if ((absX < absY) == sameSign) { 1152 oct += 1; 1153 } 1154 } 1155 1156 int wholeCount = oct << 1; 1157 memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); 1158 1159 const SkPoint* arc = &gQuadCirclePts[wholeCount]; 1160 if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) { 1161 wholeCount += 2; 1162 } 1163 pointCount = wholeCount + 1; 1164 } 1165 1166 // now handle counter-clockwise and the initial unitStart rotation 1167 SkMatrix matrix; 1168 matrix.setSinCos(uStart.fY, uStart.fX); 1169 if (dir == kCCW_SkRotationDirection) { 1170 matrix.preScale(SK_Scalar1, -SK_Scalar1); 1171 } 1172 if (userMatrix) { 1173 matrix.postConcat(*userMatrix); 1174 } 1175 matrix.mapPoints(quadPoints, pointCount); 1176 return pointCount; 1177 } 1178 1179 1180 /////////////////////////////////////////////////////////////////////////////// 1181 // 1182 // NURB representation for conics. Helpful explanations at: 1183 // 1184 // http://citeseerx.ist.psu.edu/viewdoc/ 1185 // download?doi=10.1.1.44.5740&rep=rep1&type=ps 1186 // and 1187 // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html 1188 // 1189 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) 1190 // ------------------------------------------ 1191 // ((1 - t)^2 + t^2 + 2 (1 - t) t w) 1192 // 1193 // = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} 1194 // ------------------------------------------------ 1195 // {t^2 (2 - 2 w), t (-2 + 2 w), 1} 1196 // 1197 1198 static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { 1199 SkASSERT(src); 1200 SkASSERT(t >= 0 && t <= SK_Scalar1); 1201 1202 SkScalar src2w = SkScalarMul(src[2], w); 1203 SkScalar C = src[0]; 1204 SkScalar A = src[4] - 2 * src2w + C; 1205 SkScalar B = 2 * (src2w - C); 1206 SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1207 1208 B = 2 * (w - SK_Scalar1); 1209 C = SK_Scalar1; 1210 A = -B; 1211 SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); 1212 1213 return SkScalarDiv(numer, denom); 1214 } 1215 1216 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) 1217 // 1218 // t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) 1219 // t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) 1220 // t^0 : -2 P0 w + 2 P1 w 1221 // 1222 // We disregard magnitude, so we can freely ignore the denominator of F', and 1223 // divide the numerator by 2 1224 // 1225 // coeff[0] for t^2 1226 // coeff[1] for t^1 1227 // coeff[2] for t^0 1228 // 1229 static void conic_deriv_coeff(const SkScalar src[], 1230 SkScalar w, 1231 SkScalar coeff[3]) { 1232 const SkScalar P20 = src[4] - src[0]; 1233 const SkScalar P10 = src[2] - src[0]; 1234 const SkScalar wP10 = w * P10; 1235 coeff[0] = w * P20 - P20; 1236 coeff[1] = P20 - 2 * wP10; 1237 coeff[2] = wP10; 1238 } 1239 1240 static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { 1241 SkScalar coeff[3]; 1242 conic_deriv_coeff(coord, w, coeff); 1243 return t * (t * coeff[0] + coeff[1]) + coeff[2]; 1244 } 1245 1246 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { 1247 SkScalar coeff[3]; 1248 conic_deriv_coeff(src, w, coeff); 1249 1250 SkScalar tValues[2]; 1251 int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); 1252 SkASSERT(0 == roots || 1 == roots); 1253 1254 if (1 == roots) { 1255 *t = tValues[0]; 1256 return true; 1257 } 1258 return false; 1259 } 1260 1261 struct SkP3D { 1262 SkScalar fX, fY, fZ; 1263 1264 void set(SkScalar x, SkScalar y, SkScalar z) { 1265 fX = x; fY = y; fZ = z; 1266 } 1267 1268 void projectDown(SkPoint* dst) const { 1269 dst->set(fX / fZ, fY / fZ); 1270 } 1271 }; 1272 1273 // We only interpolate one dimension at a time (the first, at +0, +3, +6). 1274 static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { 1275 SkScalar ab = SkScalarInterp(src[0], src[3], t); 1276 SkScalar bc = SkScalarInterp(src[3], src[6], t); 1277 dst[0] = ab; 1278 dst[3] = SkScalarInterp(ab, bc, t); 1279 dst[6] = bc; 1280 } 1281 1282 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { 1283 dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); 1284 dst[1].set(src[1].fX * w, src[1].fY * w, w); 1285 dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); 1286 } 1287 1288 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { 1289 SkASSERT(t >= 0 && t <= SK_Scalar1); 1290 1291 if (pt) { 1292 pt->set(conic_eval_pos(&fPts[0].fX, fW, t), 1293 conic_eval_pos(&fPts[0].fY, fW, t)); 1294 } 1295 if (tangent) { 1296 tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), 1297 conic_eval_tan(&fPts[0].fY, fW, t)); 1298 } 1299 } 1300 1301 void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { 1302 SkP3D tmp[3], tmp2[3]; 1303 1304 ratquad_mapTo3D(fPts, fW, tmp); 1305 1306 p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); 1307 p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); 1308 p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); 1309 1310 dst[0].fPts[0] = fPts[0]; 1311 tmp2[0].projectDown(&dst[0].fPts[1]); 1312 tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; 1313 tmp2[2].projectDown(&dst[1].fPts[1]); 1314 dst[1].fPts[2] = fPts[2]; 1315 1316 // to put in "standard form", where w0 and w2 are both 1, we compute the 1317 // new w1 as sqrt(w1*w1/w0*w2) 1318 // or 1319 // w1 /= sqrt(w0*w2) 1320 // 1321 // However, in our case, we know that for dst[0]: 1322 // w0 == 1, and for dst[1], w2 == 1 1323 // 1324 SkScalar root = SkScalarSqrt(tmp2[1].fZ); 1325 dst[0].fW = tmp2[0].fZ / root; 1326 dst[1].fW = tmp2[2].fZ / root; 1327 } 1328 1329 static SkScalar subdivide_w_value(SkScalar w) { 1330 return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); 1331 } 1332 1333 void SkConic::chop(SkConic dst[2]) const { 1334 SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); 1335 SkScalar p1x = fW * fPts[1].fX; 1336 SkScalar p1y = fW * fPts[1].fY; 1337 SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; 1338 SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; 1339 1340 dst[0].fPts[0] = fPts[0]; 1341 dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, 1342 (fPts[0].fY + p1y) * scale); 1343 dst[0].fPts[2].set(mx, my); 1344 1345 dst[1].fPts[0].set(mx, my); 1346 dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, 1347 (p1y + fPts[2].fY) * scale); 1348 dst[1].fPts[2] = fPts[2]; 1349 1350 dst[0].fW = dst[1].fW = subdivide_w_value(fW); 1351 } 1352 1353 /* 1354 * "High order approximation of conic sections by quadratic splines" 1355 * by Michael Floater, 1993 1356 */ 1357 #define AS_QUAD_ERROR_SETUP \ 1358 SkScalar a = fW - 1; \ 1359 SkScalar k = a / (4 * (2 + a)); \ 1360 SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ 1361 SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); 1362 1363 void SkConic::computeAsQuadError(SkVector* err) const { 1364 AS_QUAD_ERROR_SETUP 1365 err->set(x, y); 1366 } 1367 1368 bool SkConic::asQuadTol(SkScalar tol) const { 1369 AS_QUAD_ERROR_SETUP 1370 return (x * x + y * y) <= tol * tol; 1371 } 1372 1373 int SkConic::computeQuadPOW2(SkScalar tol) const { 1374 AS_QUAD_ERROR_SETUP 1375 SkScalar error = SkScalarSqrt(x * x + y * y) - tol; 1376 1377 if (error <= 0) { 1378 return 0; 1379 } 1380 uint32_t ierr = (uint32_t)error; 1381 return (34 - SkCLZ(ierr)) >> 1; 1382 } 1383 1384 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { 1385 SkASSERT(level >= 0); 1386 1387 if (0 == level) { 1388 memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); 1389 return pts + 2; 1390 } else { 1391 SkConic dst[2]; 1392 src.chop(dst); 1393 --level; 1394 pts = subdivide(dst[0], pts, level); 1395 return subdivide(dst[1], pts, level); 1396 } 1397 } 1398 1399 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { 1400 SkASSERT(pow2 >= 0); 1401 *pts = fPts[0]; 1402 SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); 1403 SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); 1404 return 1 << pow2; 1405 } 1406 1407 bool SkConic::findXExtrema(SkScalar* t) const { 1408 return conic_find_extrema(&fPts[0].fX, fW, t); 1409 } 1410 1411 bool SkConic::findYExtrema(SkScalar* t) const { 1412 return conic_find_extrema(&fPts[0].fY, fW, t); 1413 } 1414 1415 bool SkConic::chopAtXExtrema(SkConic dst[2]) const { 1416 SkScalar t; 1417 if (this->findXExtrema(&t)) { 1418 this->chopAt(t, dst); 1419 // now clean-up the middle, since we know t was meant to be at 1420 // an X-extrema 1421 SkScalar value = dst[0].fPts[2].fX; 1422 dst[0].fPts[1].fX = value; 1423 dst[1].fPts[0].fX = value; 1424 dst[1].fPts[1].fX = value; 1425 return true; 1426 } 1427 return false; 1428 } 1429 1430 bool SkConic::chopAtYExtrema(SkConic dst[2]) const { 1431 SkScalar t; 1432 if (this->findYExtrema(&t)) { 1433 this->chopAt(t, dst); 1434 // now clean-up the middle, since we know t was meant to be at 1435 // an Y-extrema 1436 SkScalar value = dst[0].fPts[2].fY; 1437 dst[0].fPts[1].fY = value; 1438 dst[1].fPts[0].fY = value; 1439 dst[1].fPts[1].fY = value; 1440 return true; 1441 } 1442 return false; 1443 } 1444 1445 void SkConic::computeTightBounds(SkRect* bounds) const { 1446 SkPoint pts[4]; 1447 pts[0] = fPts[0]; 1448 pts[1] = fPts[2]; 1449 int count = 2; 1450 1451 SkScalar t; 1452 if (this->findXExtrema(&t)) { 1453 this->evalAt(t, &pts[count++]); 1454 } 1455 if (this->findYExtrema(&t)) { 1456 this->evalAt(t, &pts[count++]); 1457 } 1458 bounds->set(pts, count); 1459 } 1460 1461 void SkConic::computeFastBounds(SkRect* bounds) const { 1462 bounds->set(fPts, 3); 1463 } 1464 1465 bool SkConic::findMaxCurvature(SkScalar* t) const { 1466 // TODO: Implement me 1467 return false; 1468 } 1469
